Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases

A multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys). To approximate the anticipated solution of the NPD equation, a linear product of variable coefficients a(i)(t) and B-i(x) B-polys has been employed. Additionally, the variable quantities in the anticipated solution are determined using the Galerkin method for minimizing errors. Before the minimization process is to take place, the NPDE is converted into an operational matrix equation which, when inverted, yields values of the undefined coefficients in the expected solution. The nonlinear terms of the NPDE are combined in the operational matrix equation using the initial guess and iterated until converged values of coefficients are obtained. A valid converged solution of NPDE is established when an appropriate degree of B-poly basis is employed, and the initial conditions are imposed on the operational matrix before the inverse is invoked. However, the accuracy of the solution depends on the number of B-polys of a certain degree expressed in multidimensional variables. Four examples of NPDE have been worked out to show the efficacy and accuracy of the two-dimensional B-poly technique. The estimated solutions of the examples are compared with the known exact solutions and an excellent agreement is found between them. In calculating the solutions of the NPD equations, the currently employed technique provides a higher-order precision compared to the finite difference method. The present technique could be readily extended to solving complex partial differential equations in multivariable problems.

Automated summary

A multivariable technique using B-polynomials is employed to estimate solutions of NPDE, with coefficients determined using the Galerkin method before conversion to an operational matrix equation. The method provides higher-order precision compared to finite difference in solving NPDE equations and has potential for solving complex partial differential equations in multivariable problems.